I have two things that I need to prove using Combinatorial proofs (where you ask a question that satisfies both sides of an equation) and I have no idea what I'm doing.
Prove $C(n,3) = C(2,2) + C(3,2) + C(4,2) + \ldots + C(n-1,2)$. The question that I need to be answered is, "How many 3 element subsets does an n element set have?" The first part is trivial, as the answer is $nC3$ by the definition of the Binomial Coefficient. However, I have no idea how to prove the second half of the equation satisfies the question. That is, how would I go about proving an $n$ element set has $C(2,2) + C(3,2) + C(4,2) + \ldots + C(n-1,2)$ subsets of 3 elements?
Prove $k C(n,k) = n C(n-1,k-1)$ combinatorial proof. While this is trivial to prove algebraically, I have to find a question that both sides of the equation answer, similar to the problem above with the 3 element subsets.
Finally, does anybody have any tips for getting through this class? The median score on our first exam was ~58% and our professor doesn't give us a curve. I got a 55, sitting at around a D- with the few homework grades we've gotten back. I am really pretty decent at other math classes, but this class has been kicking my proverbial ass. I'm just venting right here, but still. I need help.